An influenza epidemic has an infection rate modeled by I ' (t)=30 e0.04 t , where t is time
measured in days since the start of the epidemic on January 14th , and I '(t) measures the infection rate
in people per day. On January 14th there were 800 infected people on record.
a) Fully determine the function I (t), describing the total number of people infected on day t.
Remember to determine the constant c.
b) How many people became infected, to the nearest whole person, between January 14
and January 17th , that is, on the t-interval [0,3] ?
I am sure you meant
I ' (t) = 30 e^(.04t)
then I = 750 e^(.04t) + c
when t=0 , (Jan 14), I = 800
800 = 750 e^(0) + c
50 = c
a) I(t) = 750 e^(.04t) + 50
b) evaluate I(3) and I(0), then subtract the results.
To fully determine the function I(t), we need to integrate the given infection rate function I'(t).
a) To find the function I(t), we need to integrate I'(t)=30e^(0.04t) with respect to t. The indefinite integral of e^(kt) is (1/k)e^(kt). In this case, k = 0.04, so the integral will be (1/0.04)e^(0.04t).
∫I'(t)dt = ∫30e^(0.04t)dt
= (30/0.04)e^(0.04t) + C
Where C is the constant of integration.
To determine the value of C, we can use the information given that on January 14th, there were 800 infected people. We plug in t=0 into the equation and set it equal to 800.
(30/0.04)e^(0.04(0)) + C = 800
(30/0.04)(e^0) + C = 800
(30/0.04)(1) + C = 800
750 + C = 800
C = 800 - 750
C = 50
Therefore, the function I(t) describing the total number of people infected on day t is:
I(t) = (30/0.04)e^(0.04t) + 50
b) To find the number of people infected between January 14th and January 17th (t-interval [0, 3]), we need to evaluate the definite integral of I'(t) from t = 0 to t = 3.
∫[0,3] I'(t)dt = [(30/0.04)e^(0.04t)] [0,3]
= [(30/0.04)e^(0.04*3)] - [(30/0.04)e^(0.04*0)]
= (30/0.04)(e^(0.12) - 1)
Evaluating this expression will give you the number of people infected between January 14th and January 17th, rounded to the nearest whole person.